How do you write the area a of a circle as a function of its circumference?

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Answer 1 · 2015-02-13T14:18:00

We know:

Area of a circle = $A = (π) r^{2}$ $A = (pi)r^2$

Circumference of a circle = $C = 2 (π) r$ $C = 2(pi)r$

Where pi is a constant and r is the radius of the circle.

Using these two formulas we can express A in terms of C as follows:

$C^{2} = {[2 (π) r]}^{2}$ $C^2 = [2(pi)r]^2$

$\Rightarrow C^{2} = 4 [{(π)}^{2}] r^{2}$ $=> C^2 = 4[(pi)^2]r^2$

$\Rightarrow C^{2} = 4 (π) [(π) r^{2}]$ $=> C^2 = 4(pi)[(pi)r^2]$

As $(π) r^{2} = A$ $(pi)r^2 = A$

$\Rightarrow C^{2} = 4 (π) A$ $=> C^2 = 4(pi)A$

Therefore:

$A = \frac{C^{2}}{4 (π)}$ $A = (C^2)/[4(pi)]$